Question

Find the equation of the plane which passes through the points (0, 0, 0) and (3, –1, 2) and is parallel to the line $fraction numerator straight x minus 4 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator negative 4 end fraction space equals space fraction numerator straight z plus 1 over denominator 7 end fraction.$

Solution

The equation of any plane through (0, 0, 0) is
A (x – 0) + B (y – 0) + C (z – 0) = 0
or A x + B y + C z = 0 ...(1)
∴ it passes through (3, –1, 2)
∴ 3A – B + 2C = 0 ...(2)
Since plane (1) is parallel to the line $fraction numerator straight x minus 4 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator negative 4 end fraction space equals space fraction numerator straight z plus 1 over denominator 7 end fraction.$

∴ normal to the plane with direction ratios A, B, C is perpendicular to the line with direction ratios 1, – 4, 7.
∴ A (1) + B (– 4) + C(7) = 0     [∴ a₁a₂ +b₁b₂ +c₁c₂= 0]
∴ A – 4B + 7C = 0    ...(3)
Solving (2) and (3), we get,
   $fraction numerator straight A over denominator negative 7 plus 8 end fraction space equals space fraction numerator straight B over denominator 2 minus 21 end fraction space equals space fraction numerator straight C over denominator negative 12 plus 1 end fraction$
$therefore space space space space straight A over 1 space equals space fraction numerator straight B over denominator negative 19 end fraction space equals space fraction numerator straight C over denominator negative 11 end fraction space equals space straight k space left parenthesis say right parenthesis therefore space space space space space straight A space equals space straight k comma space space space straight B space equals space minus 19 space straight k comma space space space straight C space equals space minus 11 space straight k$
Putting values of A, B, C in (1), we get,
k x – 19 k y – 11 k z = 0
or x – 19 y – 11 z = 0
which is required equation of plane.

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Three Dimensional Geometry

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